Theorem rmoeqd 2676

Description: Equality deduction for restricted at-most-one quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
raleqd.1	|- ( A = B -> ( ph <-> ps ) )

Assertion

Ref	Expression
rmoeqd	|- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) )

Distinct variable groups:   x, A   x, B

Allowed substitution hints:   ph( x)   ps( x)

Proof of Theorem rmoeqd

Step	Hyp	Ref	Expression
1		rmoeq1 2668	. 2 |- ( A = B -> ( E* x e. A ph <-> E* x e. B ph ) )
2		raleqd.1	. . 3 |- ( A = B -> ( ph <-> ps ) )
3	2	rmobidv 2658	. 2 |- ( A = B -> ( E* x e. B ph <-> E* x e. B ps ) )
4	1, 3	bitrd 187	1 |- ( A = B -> ( E* x e. A ph <-> E* x e. B ps ) )

Syntax hints:   -> wi 4   <-> wb 104   = wceq 1348   E*wrmo 2451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-cleq 2163  df-clel 2166  df-nfc 2301  df-rmo 2456

This theorem is referenced by: (None)